Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1805 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,12)(8,10)(9,11)(13,14,18,20)(15,25,29,37)(16,30,33,39)(17,24,19,23)(21,28,35,26)(22,34,36,31)(27,38,32,40)(41,42,44,43);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,20)(15,26)(16,31)(17,19)(21,37)(22,39)(25,35)(28,29)(30,36)(33,34)(38,40)(42,43);(13,15,18,29)(14,21,20,35)(16,32,33,27)(17,26,19,28)(22,40,36,38)(23,25,24,37)(30,44,39,41)(31,43,34,42);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,33)(14,22,20,36)(15,27,29,32)(17,31,19,34)(21,38,35,40)(23,30,24,39)(25,41,37,44)(26,42,28,43);(6,11)(10,12)(13,17,18,19)(14,23,20,24)(15,28,29,26)(16,31,33,34)(21,37,35,25)(22,30,36,39)(27,43,32,42)(38,44,40,41)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425325);